General Results for Dislocation Type Equations

arXiv:math/0702153v1[math.AP]6Feb2007GENERALRESULTSFORDISLOCATIONTYPEEQUATIONSGUYBARLES,PIERRECARDALIAGUET,OLIVIERLEY&R´EGISMONNEAUAbstract.WeareinterestedinnonlocalEikonalEquationsaris-inginthestudyofthedynamicsofdislocationslinesincrystals.Forthesenonlocalbutalsononmonotoneequations,onlytheexis-tenceanduniquenessofLipschitzandlocal-in-timesolutionswereavailableinsomeparticularcases.Inthispaper,weproposeadefi-nitionofweaksolutionsforwhichweareabletoprovetheexistenceforalltime.Thenwediscusstheuniquenessofsuchsolutionsinseveralsituations,bothinthemonotoneandnonmonotonecase.1.IntroductionInthisarticleweareinterestedinthedynamicsofdefectsincrys-tals,calleddislocations.Thesedislocationsarelinesmovingintheirslipplane.TheirdynamicsisgivenbyanormalvelocityproportionaltothePeach-Koehlerforcewhichisanonlocalfunctionoftheshapeofthedislocationlineitself.Moreprecisely,if,attimet,thedislocationlineistheboundaryofanopensetΩt⊂RNwithN=2forthephysicalapplication,thenormalvelocitytothesetΩtisgivenby(1)Vn=c0⋆11Ωt+c1where11Ωt(x)istheindicatorfunctionofthesetΩt,whichisequalto1ifx∈Ωtandequalto0otherwise.Thefunctionc0(x,t)isakernelwhichonlydependsonthephysicalpropertiesofthecrystalandontheBurgersvectorassociatedtothedislocationline.Inthespecialcaseofapplicationtodislocations,thekernelc0doesnotdependontime,1991MathematicsSubjectClassification.49L25,35F25,35A05,35D05,35B50,45G10.Keywordsandphrases.NonlocalHamilton-JacobiEquations,dislocationdynamics,nonlocalfrontpropagation,level-setapproach,geometricalprop-erties,lower-boundgradientestimate,viscositysolutions,eikonalequation,L1−dependenceintime.12G.BARLES,P.CARDALIAGUET,O.LEYANDR.MONNEAUbuttokeepageneralsettingweallowhereadependenceonthetimevariable.Here⋆denotestheconvolutionisspace,namely(2)(c0(·,t)⋆11Ωt)(x)=ZRNc0(x−y,t)11Ωt(y)dy,andc1(x,t)isanadditionalcontributiontothevelocity,createdbyexteriorstressappliedonthematerial.WerefertoAlvarezetal.[3],andRodneyetal.[24]forapresentationofthismodel.Althoughequation(1)seemsverysimple,thereareonlyafewknownresults.Undersuitableassumptionsontheinitialdataandonc0,c1,theexistenceanduniquenessofthesolutionisknownintwoparticularcases:eitherforshorttime(see[3]),orforalltimeundertheaddi-tionalassumptionthatVn≥0,whichisforinstancealwayssatisfiedforc1satisfyingc1(x,t)≥|c0(·,t)|L1(RN)(see[2],[12]or[5]foralevelsetformulation).Inthegeneralcase,eventheexistenceforalltimeofsolutionstoequation(1)isnotknown.Thedifficultyisduetothefactthatingeneralthekernelc0hasnegativevalues.Itfollowsthatthefrontpropagationproblem(1)doesnotstatisfyanypreservationofinclu-sionproperty.Nevertheless,alevel-settypeequationcanbederivedandisknowntobewell-posedinthecaseofclassicalnormalvelocities(constantnormalvelocities,motionbymeancurvatureandevennon-localvelocitieswhichhavetherightmonotonicitypropertiesleadingtoinclusionprinciples.SeeforinstanceGiga’smonograph[18]).Afurtherconsequenceofthisbadsignofthekernel,isthatitpreventsfromusingclassicaltechniquesofhalf-relaxedlimitswhenbuildingap-proximatesolutionsof(1).Thebestwecouldobtainisthatthelimitsareonlyweaksolutions.Theaimofthispaperistodescribeageneralapproachofthesedislocations’dynamics,basedonthelevel-setapproach,whichallowsustointroduceasuitablenotionofweaksolutions,toprovetheexistenceoftheseweaksolutionsforalltimeandtoanalysetheuniqueness(ornon-uniqueness)ofthesesolutions.1.1.Weaksolutionsofthedislocationequation.Weproceedasinthelevel-setapproachtoderiveanequationforthedislocationline.WereplacetheevolutionofasetΩt(thestrongsolution),bytheevo-lutionofafunctionusuchthatΩt={u(·,t)0}.RoughlyspeakingthedislocationlineisrepresentedbythezerolevelsetofthefunctionGENERALRESULTSFORDISLOCATIONTYPEEQUATIONS3uwhichsolvesthefollowingequation(3)(∂u∂t=(c0(·,t)⋆11{u(·,t)≥0}(x)+c1(x,t))|Du|inRN×(0,T)u(·,0)=u0inRN,where(2)nowreadsc0(·,t)⋆11{u(·,t)≥0}(x)=ZRNc0(x−y,t)11{u(·,t)≥0}(y)dy.(4)Notethat(3)isnotreallyalevel-setequationsinceitisnotinvariantundernondecreasingchangesoffunctionsu→ϕ(u)whereϕisnon-decreasing.AsnoticedbySlepˇcev[25],thenaturallevel-setequationshouldbe(11),seeSection1.2.Weintroducethefollowingdefinitionofweaksolutions,whichusesitselfthedefinitionofL1-viscositysolutions,recalledinAppendixA.Definition1.1.(Classicalandweaksolutions)ForanyT0,wesaythatafunctionu∈W1,∞(RN×[0,T))isaweaksolutionofequation(3)onthetimeinterval[0,T),ifthereissomemeasurablemapχ:RN×(0,T)→[0,1]suchthatuisaL1-viscositysolutionof(5)(∂u∂t=¯c(x,t)|Du|inRN×(0,T)u(·,0)=u0inRN,where(6)¯c(x,t)=c0(·,t)⋆χ(·,t)(x)+c1(x,t)and(7)11{u(·,t)0}(x)≤χ(x,t)≤11{u(·,t)≥0}(x),foralmostall(x,t)∈RN×[0,T].Wesaythatuisaclassicalsolutionofequation(3)ifuisaweaksolutionto(5)andif(8)11{u(·,t)0}(x)=11{u(·,t)≥0}(x)foralmostall(x,t)∈RN×[0,T].Notethatwehaveχ(x,t)=11{u(·,t)0}(x)=11{u(·,t)≥0}(x)foralmostall(x,t)∈RN×[0,T]forclassicalsolutions.Tostateourfirstexistenceresult,weusethefollowingassumptions(H0)u0∈W1,∞(RN),−1≤u0≤1andthereexistsR00suchthatu0(x)≡−1for|x|≥R0,4G.BARLES,P.CARDALIAGUET,O.LEYANDR.MONNEAU(H1)c0∈C([0,T);L1RN
),Dxc0∈L∞([0,T);L1RN
),c1∈C(RN×[0,T))andthereexistsconstantsM1,L1suchth